Given triangle ABC. If a point was placed inside thatformed a line segment with

Question

Given triangle ABC. If a point was placed inside thatformed a line segment with each vertices, where should that pointbe placed to minimize the total sum of distances from that point tothe vertices?
I have used the geogebra program, and I can tell you it is notthe circumcenter of the triangle. I can not figure thisproblem out. I believe it is different for right, acute, andobtuse triangles. Can someone please help me with somegraphical evidence? I will rate lifesaver Given triangle ABC. If a point was placed inside thatformed a line segment with each vertices, where should that pointbe placed to minimize the total sum of distances from that point tothe vertices?
I have used the geogebra program, and I can tell you it is notthe circumcenter of the triangle. I can not figure thisproblem out. I believe it is different for right, acute, andobtuse triangles. Can someone please help me with somegraphical evidence? I will rate lifesaver

Explanation / Answer

The point is characterized by the fact that the segments fromthe point to the three vertices make 120 degree angles witheach other. I think this will construct the required point: Construct an equilateral triangle on of a side of thetriangle, with the equilateral triangle lying outside the triangle.Bisect two of the angles and determine their point of intersection.Construct a circle using the intersection point as the center andradius equal to the distance to the endpoints of theside. Repeat the above on a second side of the triangle. The twocircles will intersect in the triangle at the desired point.

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